Locate the reset button on either the burner motor or the stack, depending on your furnace design.
Push the reset button one time. This releases a small amount of fuel to the burner. Modern furnaces use spark ignition to light the fuel, rather than a pilot light. Your furnace has a sensor that indicates when fuel is in the burner and it releases a spark automatically. This should light the fuel.
Push the reset button a second time if the spark did not ignite it. If the oil tank ran dry, then you may need to prime the burner more than once. Do not press it more than two times, or there will be too much fuel in the burner, similar to flooding the carburetor in a vehicle. If this second reset has not lit the fuel, then proceed to the next step.
Check the burner nozzle for dirt. The flame down in the burner passes up through the nozzle and heats the air as it passes by. Additionally, the flame feeds on oxygen in the air. A clogged nozzle prevents oxygen from reaching the flame, so it will not light. Changing the burner nozzle should solve this problem.
Check the flame sensor if the reset button lights the fuel but the flame quickly goes out again. Cleaning the dirt and soot from the sensor should restore it to good working order.
Replace the burner motor if it has burned out. The motor is a sealed unit, so you cannot repair it. Sometimes the problem is with the on-off switch, not the motor. Once the switch or motor is replaced, the flame will light when you push the reset button.